CSE%20246:%20Computer%20Arithmetic%20Algorithms%20and%20Hardware%20Design

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CSE 246 Computer Arithmetic Algorithms and
Hardware Design
Fall 2006 Lecture 9 Floating Point Numbers

• Instructor
• Prof. Chung-Kuan Cheng

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Motivation

• Maximal information with given bit numbers.
• Arithmetic with proper precision.
• Fairness of rounding.
• Features at the expenses of the complexity of the
  operations.

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Topics

• Floating Point Numbers (IEEE P754)
• Standard
• Operations
• Exceptional Situations
• Rounding Modes
• Numerical Computing with IEEE Floating Point
  Arithmetic, Michael L. Overton, SIAM

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Standard
232 ? Typically

• Goal Dynamic Range
• largest / smallest
• If too large, holes between s

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Standard

• ulp (unit in the last place)
• Difference between two consecutive values of the
  significand.
• 3 Parts? x s besign, significand, exponent

Sign Bit
23-bit Significand
8-bit exponent
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Standard

• e1e2e3e4e5e6e7e8s1s2s3s22s23
• 1.s1s2s3s22s23 normalized number
• 0.s1s2s3s22s23 denormalized number
• e1e2e3e4e5e6e7e8
• 0 0 0 0 0 0 0 0 0 x0.s1s2s3s22s23 2-126
• 1 0 0 0 0 0 0 0 1 x1.s1s2s3s22s23
  2-126
• 2 0 0 0 0 0 0 1 0 x1.s1s2s3s22s23
  2-125
• .
• 127 0 1 1 1 1 1 1 1 x1.s1s2s3s22s23 20
• .
• 253 1 1 1 1 1 1 0 1 x1.s1s2s3s22s23
  2126
• 1 1 1 1 1 1 1 0 x1.s1s2s3s22s23 2127
• 1 1 1 1 1 1 1 1 x Inf if (s1 s23) 0,
  NaN otherwise. NaN ? Not a Number

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Standard

• 0.01x2-3 0.001x2-2
• Same number, so normalize to remove redundancy
• Use a default 1 in front for one more bit
  precision.
• Smallest Number
• 0.0001x2-126 1.0x2-23x2-126
• 1x2-149

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Standard - Example

• eeeeeeee sssss sssss sssss sssss sss
• 0 00000000 00000000000000000000000
  0.0000x2-126
• 1 00000000 00000000000000000000000
  -0.0000x2-126
• 0 00000000 00000000000000000000001
  0.0001x2-149
• 0 00000001 00000000000000000000000
  1.0000x2-126
• normalized minimum
• 0 00000001 00000000000000000000001
  1.0001x2-126
• .
• .
• 0 01111111 00000000000000000000000 1.0000x20
• 0 01111111 00000000000000000000001 1.0001x20
• 0 10000000 00000000000000000000001 1.0001x21

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Standard Example Cont.

• 0 11111110 00000000000000000000000 1.0000x2127
• 0 11111110 00000000000000000000001 1.0001x2127
• 0 11111110 11111111111111111111111 1.1111x2127
• - Normalized Maximum
• 0 11111111 00000000000000000000000 Inf
• Nmin 1.0 x 2-126
• Nmax (2 2-23)2127

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Double Floating Point

• e1e2e11 s1s2s52
• 0 00000 s1s2s52 x0.s1s2s52 2-1022
• 0 00001 s1s2s52 x1.s1s2s52 2-1022
• .
• .
• 0 01111 s1s2s52 x1.s1s2s52 20
• 0 10000 s1s2s52 x1.s1s2s52 21
• .
• .
• 0 11110 s1s2s52 x1.s1s2s52 21023
• 0 11111 s1s2s52 xInf if (s1s52)0

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Overflow/Underflow
Underflow
Sparser
Denser
Overflow
Nmin
Nmax
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Addition/Multiplication

• s1xbe1 (s2xbe2) sxbe
• s1xbe1 s2/be1-e2 x be1
• (s1 s2/be1-e2) x be1
• (s1xbe1) x (s2xbe2) (s1xs2)be1e2

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Exceptions

• a/0 Inf if a gt 0
• a/Inf 0 if a ! 0
• a0 0
• aInf Inf if a gt 0
• a Inf Inf
• 0Inf invalid operation (NaN)
• 0/0 invalid operation (NaN)
• Inf - Inf NaN
• NaP op a NaN

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Rounding Mode

• Adder Output Cout z1z0.z-1z-2z-l GRS
• Guard Bit
• Round Bit
• Sticky Bit, OR of all bits below bit R
• 1.101 x 23
• 1.110 x 23
• 11.011 x 23
• 1.1011x24 Normalize need to round or

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Rouding

• 1.110 23
• - 1.101 23
• 0.001 23
• 1.000 20 normalize
• 1.101 23
• - 1.111 22
• 1.101 23
• - 0.1101 23
• 0.1101 23
• 1.101 22

Guard bit
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Rounding

• Round to the nearest even
• 1.10111
• toward 0 1.1011
• Toward Inf 1.1100
• Toward -Inf 1.1011

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Conventional Rounding Error

• Rounding Error
• 1.10100 ? 1.101 0
• 1.10101 ? 1.101 -0.25
• 1.10110 ? 1.110 0.5
• 1.10111 ? 1.110 0.25
• Average Error 0.5/4 0.125